
where \(\mathcal {F}\)  is the data-driven, sparsifying, analysis operator and \(p\)  is the number of sparsity-promoting regularizers. The convex and non-convex regularizers that are considered in our work are the \(\ell _1\) -norm, minimax-concave penalty (MCP) [1]}, and smoothly clipped absolute deviation (SCAD) [2]}. While the \(\ell _1\) -norm is convex, the other two penalties – MCP and SCAD – are not convex. The MCP and SCAD penalties approximate the \(\ell _0\)  pseudonorm better than the \(\ell _1\)  norm and hence give rise to better recovery performance [1]}. The ideal sparse recovery problem requires minimization of the \(\ell _0\)  pseudonorm, but it is NP-hard and hence tractable solvers that approximate \(\ell _0\)  minimization are preferred. The MCP and SCAD penalties perfectly fit the bill and hence we consider them for composite regularization. The three penalties under consideration and the corresponding proximal operators are given in Table REF .
